0 electrodynamics solutions

As mentioned earlier, the theory of optical absorption of small particles was proposed by Mie in 1908 [6[. Mie s electrodynamic solution to the problem of light interacting with particles involved solving Maxwell s equations with appropriate boundary conditions and leads to a series of multipole oscillations for the extinction (Cext) and scattering (Cgca) cross-sections of the nanoparticles. Thus... 

Finally, the much deeper and difficult questions concerned with the consistency of commutation rules and of the equations of motion, and the questions concerned with the existence of solutions have not been raised in these chapters. They have also in fact not received even a partial answer in the literature.- The question of the limits of quantum electrodynamics may, however, receive an experimental answer in the foreseeable future. 

These speculations about the ionic, polar, or electronic nature of chemical bonding, which arose largely from solution theory, resulted mostly in static models of the chemical bond or atom structure. In contrast is another tradition, which is more closely identified with ether theory and electrodynamics. This tradition, too, may be associated with Helmholtz, especially by way of his contributions to nineteenth-century theories of a "vortex atom" that would explain chemical affinities as well as the origin of electromagnetism, radiation, and spectral lines. 

Similarly, when the electromagnetic signals that constitute a set of orthogonally coupled Electric (E) and Magnetic field (B) vectors are introduced inside a geometrical boundary, which in the case in a cylindrical resonant cavity, electrodynamics comes into the play. Solutions to the Maxwell s equations according to the... 

All of these solutions fail for Pe = 0(1), so Davis and his colleagues (Zhang and Davis, 1987 Taflin and Davis, 1987) performed evaporative mass transfer experiments over the Peclet number range 0.01 < Pe < 4. using electrodynamically levitated droplets of hexadecane in flowing Nj and He and dodecanol in Nj. 

In attempts to develop conventional electrodynamic models of the individual photon, it is difficult to finding axisymmetric solutions that both converge at the photon center and vanish at infinity. This was already realized by Thomson [22] and later by other investigators [23]. 

The absence of an E(3) field does not affect Lorentz symmetry, because in free space, the field equations of both 0(3) electrodynamics are Lorentz-invariant, so their solutions are also Lorentz-invariant. This conclusion follows from the Jacobi identity (30), which is an identity for all group symmetries. The right-hand side is zero, and so the left-hand side is zero and invariant under the general Lorentz transformation [6], consisting of boosts, rotations, and space-time translations. It follows that the B<3) field in free space Lorentz-invariant, and also that the definition (38) is invariant. The E(3) field is zero and is also invariant thus, B(3) is the same for all observers and E(3) is zero for all observers. 

The E(3) field is zero in frame K, and a Z boost means [from Eq. (403)] that it is zero in frame K. This is consistent with the fact that Ea> is a solution of an invariant equation, the Jacobi identity (30) of 0(3) electrodynamics. Finally, we can consider two further illustrative example boosts of El ]] in the X and Y directions, which both produce the following result ... 

In conclusion, the homogeneous field equation of 0(3) electrodynamics is Lorentz-invariant, and all its classical solutions must be also Lorentz-invariant. The same result is obtained therefore in QED. 

More than a century later, Lehnert [7] introduced and developed [7-10] the concept of vacuum charge on the classical level, and showed [7-10] that this concept leads to advantages over the Maxwell-Heaviside equations in the description of empirical data, for example, the problem of an interface with a vacuum [7-10,15]. The introduction of a vacuum charge leads to axisymmetric vacuum solutions akin to the B(3> vacuum component of 0(3) electrodynamics... 

By way of introduction to the Noether currents and charges that exist in 0(3) electrodynamics, the inhomogeneous field of Eq. (32) can be considered in the vacuum (source-free space) and split into two particular solutions ... 

In this section, the field equations (31) and (32) are considered in free space and reduced to a form suitable for computation to give the most general solutions for the vector potentials in the vacuum in 0(3) electrodynamics. This procedure shows that Eqs. (86) and (87) are true in general, and are not just particular solutions. On the 0(3) level, therefore, there exist no topological monopoles or magnetic charges. This is consistent with empirical data—no magnetic monopoles of any kind have been observed in nature. 

In this final section, it is shown that the three magnetic field components of electromagnetic radiation in 0(3) electrodynamics are Beltrami vector fields, illustrating the fact that conventional Maxwell-Heaviside electrodynamics are incomplete. Therefore Beltrami electrodynamics can be regarded as foundational, structuring the vacuum fields of nature, and extending the point of view of Heaviside, who reduced the original Maxwell equations to their presently accepted textbook form. In this section, transverse plane waves are shown to be solenoidal, complex lamellar, and Beltrami, and to obey the Beltrami equation, of which B is an identically nonzero solution. In the Beltrami electrodynamics, therefore, the existence of the transverse 1 = implies that of , as in 0(3) electrodynamics. 

This argument shows again that Maxwell-Heaviside electrodynamics is incomplete, because B(3) is zero. General solutions are given in this section of the Beltrami equation, which is an equation of 0(3) electrodynamics. Therefore these solutions are also general solutions of 0(3) electrodynamics in the vacuum. 

The Bii] component [which is nonzero only on the 0(3) level] is a solution of the Beltrami equation (885) with k = 0. Therefore, in Beltrami electrodynamics, Bii] is a solenoidal, irrotational, complex lamellar and Beltrami field in the vacuum, and is also a propagating field. The B 1 component in Beltrami... 

Without the additional 3-symmetry condition, the resulting Whittaker 4-symmetry EM energy flow mechanism resolves the nagging problem of the source charge concept in classical electrodynamics theory. Quoting Sen [10] The connection between the field and its source has always been and still is the most difficult problem in classical and quantum mechanics. We give the solution to the problem of the source charge in classical electrodynamics. 

It appears that a permanent solution to the world energy problem, dramatic reduction of biospheric hydrocarbon combustion pollution, and eliminating the need for nuclear power plants (whose nuclear component is used only as a heater) could be readily accomplished by the scientific community [18]. However, to solve the energy problem, we must (1) update the century-old false notions in electrodynamic theory of how an electrical circuit is powered and (2) correct the classical electrodynamics model for numerous foundations flaws. 

We use the foregoing hypothesis to propose a solution to a previously unsolved major foundations problem in electrodynamics. Quoting Sen [10] ... 

Eerily, our scientific community ignores the terrible > 135-year-old foundation errors in classical electromagnetics and assures us that this is the best that electrodynamics can do. In fact, the scientific community has not yet even recognized the problem, much less the solution. As Bunge [40] so poignantly stated in 1967 it is not usually acknowledged that electrodynamics, both classical and quantal, are in a sad state. ... 

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